1

Complex Dynamics of Blackouts in Power Transmission Systems

B. A. Carreras and V. E. Lynch

Oak Ridge National Laboratory, Oak Ridge, TN 37831

I. Dobson

ECE Dept., University of Wisconsin, Madison, WI 53706 USA

D. E. Newman

Physics Dept., University of Alaska, Fairbanks, AK 99775 USA

Abstract

In order to study the complex global dynamics of a series of blackouts in power

transmission systems a dynamical model of such a system has been developed. This

model includes a simple representation of the dynamical evolution by incorporating the

growth of power demand, the engineering response to system failures, and the upgrade of

generator capacity. Two types of blackouts have been identified, each having different

dynamical properties. One type of blackout involves the loss of load due to transmission

lines reaching their load limits but no line outages. The second type of blackout is

associated with multiple line outages. The dominance of one type of blackout over the

other depends on operational conditions and the proximity of the system to one of its two

critical points. The model displays characteristics such as a probability distribution of

blackout sizes with power tails similar to that observed in real blackout data from North

America.

2

Power transmission systems are complex systems that evolve over years in

response to the economic growth of the country and to continuously increasing

power demand. In spite of the reliability of these systems, there are widespread

disturbances that have significant cost to society. The average frequency of

blackouts in the United States is about one every 13 days. This frequency has not

changed over the last 30 years. Also the probability distribution of blackout sizes

has a power tail; this dependence indicates that the probability of large blackouts is

relatively high. Indeed, although large blackouts are rarer than small blackouts, it

can be argued that their higher societal cost makes the risk of large blackouts

comparable to or exceed the risk of small blackouts.

The operation of power transmission systems is studied from the perspective of

complex dynamics in which a diversity of opposing forces regulate both the

maximum capabilities of the system components and the loadings at which they

operate. These forces enter in a nonlinear manner and may cause an evolution

process to be ultimately responsible for the regulation of the system. This view of a

power transmission system considers not only the engineering and physical aspects

of the power system, but also the economic, regulatory, and political responses to

blackouts and increases in load power demand. From this perspective, the search

for the cause of the blackouts must not be limited to the trigger of the blackout,

which is normally a random event, but it must also consider the dynamical state of

the power transmission system. A detailed incorporation of all these aspects of the

dynamics into a single model is extremely complicated. Here, a simplified model is

discussed with some approximate overall representation of the opposing forces

controlling the system dynamics. This model reproduces some of the main features

of North American blackout data.

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I. Introduction

Power transmission systems are complex systems that evolve over years in response

to the economic growth of the country and to continuously increasing power demand.

The evolution and reliability of these systems are leading engineering accomplishments

of the last century that underpin developed societies. Nevertheless, widespread

disturbances of power transmission systems that have significant cost to society are

consistently present. An analysis of blackouts1 done in the 1970s indicated that the

average frequency of blackouts in the United States was one every 14 days. More recent

analyses2,3 of 15 years of North American Electrical Reliability Council (NERC) data on

blackouts of the North American power grid4 gave an average frequency of blackouts of

one every 13 days. Furthermore, these analyses show that the distribution of blackout

sizes has a power tail with an exponent of about –1.3±0.2. These results indicate that the

probability of large blackouts is relatively high. Indeed, although large blackouts are

rarer than small blackouts, it can be argued that combining their higher societal costs with

their relatively high probability makes the risk of large blackouts comparable to or

greater than the risk of small blackouts.5

It is clear that individual blackouts are triggered by random events ranging from

equipment failures and bad weather to vandalism.4 The blackouts then typically become

widespread through a series of cascading events. However, it must be remembered that

these individual blackouts occur in a power transmission system that is itself slowly and

dynamically evolving in its design, configuration and operation. For example, the

loading of system components relative to their maximum loading is a key factor

governing the propagation of component failures and this loading evolves as the system

components or operational policies are upgraded. The existence of a power tail in the

distribution of blackouts and the long time correlations seen in the system suggests that

underlying the large-scale blackouts may be a dynamically caused proximity to a critical

point. It should be noted that the size of a given blackout is unrelated to the particular

triggering event that initiated that blackout.

To investigate such a possibility, we propose a model for power transmission

systems6,7 that involves not only the dynamics of the generator dispatch but also the

4

evolution of the system under a continuous increase in demand. This model shows how

the slow opposing forces of load growth and network upgrades in response to blackouts

could self-organize the power system to a dynamic equilibrium. Blackouts are modeled

by overloads and outages of transmission lines determined in the context of linear

programming (LP) dispatch of a DC load flow model. This model shows complex

dynamical behaviors and has a variety of transition points as a function of increasing

power demand.8 Some of these transition points have the characteristic properties of a

critical transition. That is, when the power demand is close to a critical value, the

probability distribution function (PDF) of the blackout size has an algebraic tail, and the

system changes sharply across the critical point. Because of this, the risk of a global

blackout increases sharply at the critical transition.

The fact that, on one hand, there are critical points with maximum power flow

through the network and, on the other hand, there is a self-organization process that tries

to maximize efficiency and minimize risk, may lead to a power transmission model

governed by self-organized criticality (SOC).9 Such a possibility was first explored with

a simple cellular automaton model10 that incorporates neither the circuit equations nor the

type of long-term dynamics discussed above. In this paper, we study the dynamical

properties of a power transmission model6,7 that does incorporate these two components.

There have been some other complex system approaches to modeling aspects of power

system blackouts. In the most closely related work, Chen and Thorp11,12 modeled power

system blackouts using DC load flow and LP dispatch and represented in detail hidden

failures of the protection system. They obtained the distribution of power system

blackout size by rare event sampling, and studied blackout risk assessment and mitigation

methods. Stubna and Fowler13 applied a modified "Highly Optimized Tolerance" (HOT)

model to fit North American blackout data for blackout sizes measured by both power

shed and customers disconnected. Using a different approach, Roy, Asavathiratham,

Lesieutre, and Verghese constructed randomly generated tree networks that abstractly

represent influences between idealized components.14 In that work, components can be

failed or operational according to a Markov model that represents both internal

component failure and repair processes and influences between components that cause

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failure propagation. The effects of the network degree of connectivity and

intercomponent influences on the failure size and duration were studied. Similarly,

Pepyne et al.15 used a Markov model for discrete-state power system nodal components

but had failures propagate along the transmission lines of a power system network with a

fixed probability. DeMarco16 and Parrilo et al.17 addressed the challenging problem of

determining cascading failure due to dynamic transients by using hybrid nonlinear

differential equation models. DeMarco used Lyapunov methods applied to a smoothed

model; Parrilo et al. used Karhunen- Loeve and Galerkin model reduction to address the

problem.

The rest of this paper is organized as follows. In Sect. 2, we introduce a dynamical

model of power transmission system evolution over long time scales. Details of the

power flow model and of the fast time scale dynamics are provided in Appendix A. In

Sect. 3, numerical results of the model are reported with an analysis of the time and space

correlations introduced by the dynamics. In Sect. 4, we analyze the effect of changing

the ratio of generator capacity margin to maximum load fluctuation. This ratio allows the

separation of the dynamics into two different regimes. The conclusions are given in

Sect. 5.

2. Dynamical model for power transmission

In modeling the dynamics of power transmission systems, one must consider two

intrinsic time scales. There is a slow time scale, of the order of days to years, over which

load power demand slowly increases. Over this time scale, the network is upgraded in

engineering responses to blackouts and in providing more generator power in response to

demand. As we shall see, these slow opposing forces of load–increase and

network–upgrade self-organize the system to a dynamic equilibrium. The dynamical

properties of this model are the main topic of this paper. In power transmission systems,

there is also a fast time scale, of the order of minutes to hours, over which power is

dispatched through the network within which (depending on the conditions of the

network) cascading overloads or outages may lead to a blackout.

Over the fast time scale, we solve the standard DC power flow equation for a given

distribution of load demand. We use the standard LP method18–20 with the usual

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constraints on generating power capability and transmission line limits to solve the

generator power dispatch. An example of a power transmission network used in these

studies is the IEEE 118 bus network21 shown in Fig. 1. Details of the fast dynamics can

be found in Refs. 6 and 7 and a summary description is given in Appendix A.

In any network, the network nodes (buses) are either loads (L) (gray squares in

Fig. 1), or generators (G), (black squares in Fig. 1). The power Pi injected at each node is

positive for generators and negative for loads, and the maximum power injected is Pimax .

The transmission line connecting nodes i and j has power flow Fij, maximum power flow

Fijmax , and impedance zij. There are Nl lines and N N NN G L= + total nodes, where NG is

the number of generators and NL is the number of loads.

The slow dynamics proposed in Refs. 6 and 7 has three components: (1) the growth of

the demand, (2) response to blackouts by upgrades in the grid transmission capability and

(3) response to increased demand by increasing maximum generator power. These

components of the model are translated into a set of simple rules. We simplify the time

scale by regarding one blackout to be possible each day at the peak loading of that day.

At the beginning of the day t, we apply the following rules:

1. The demand for power grows. All loads are multiplied by a fixed parameter l that

represents the daily rate of increase in electricity demand. On the basis of past

electricity consumption in the United States, we estimate that l = 1.00005. This value

corresponds to a yearly rate of 1.8%.

P t P t i Li i( ) = -( ) Œl 1 for . (1)

To represent the daily local fluctuations in power demand, all power loads are

multiplied by a random number r, such that 2 - £ £g gr , with 1 £ g £ 2. The power

transmission grid is improved. We assume a gradual improvement in the

transmission capacity of the grid in response to outages and blackouts. This

improvement is implemented through an increase of Fijmax for the lines that have

overloaded during a blackout. That is,

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F t F tij ijmax max( ) = -( )m 1 , (2)

if the line ij overloads during a blackout. We take m to be a constant greater than 1

and in the present studies we have varied m in the range 1.01 £ m £ 1.1.

It is customary for utility engineers to make prodigious efforts to avoid blackouts,

and especially to avoid repeated blackouts with similar causes, which we have

simplified into this one parameter m. In general, these responses to blackouts occur

on a range of time scales longer than one day. Responses include repair of damaged

equipment, more frequent maintenance, changes in operating policy away from the

specific conditions causing the blackout, installing new equipment to increase system

capacity, and adjusting or adding system alarms or controls. The responses reduce

the probability of events in components related to the blackout, either by lowering

their probabilities directly or by reducing component loading by increasing

component capacity or by transferring some of the loading to other components. The

responses are directed toward the components involved in causing the blackout. Thus

the probability of a similar blackout occurring is reduced; at least until load growth

degrades the improvements that were made. There are similar but less intense

responses to unrealized threats to system security, such as near misses and simulated

blackouts.

By simplifying all engineering responses into a single parameter m we crudely

represent all these responses to a blackout. The response is modeled as happening on

the next day, but the effect is eventually cancelled by the slow load increase. Because

of the disparity between these two time scales, at this level of modeling it does not

seem crucial to have an accurate estimate of the response time, and the one-day time

scale may be reasonable.

3. The maximum generator power is increased in response to the load demand as

follows:

(a) The increase in power is quantized. This can reflect either the upgrade of a power

plant or the addition of generators. The increase is taken to be a fixed ratio to the

total power. Therefore, we introduce the quantity

8

DP P Na T G∫ ( )k , (3)

where PT is the total power demand, NG the number of generators, and k is a

parameter that we have taken to be a few percent.

(b) To be able to increase the maximum power in node j, the sum of the power flow

limits of the lines connected to j should be larger than the existing generating

power plus the addition at node j. This requirement maintains the coordination of

the maximum generator power ratings with the line ratings.

(c) A second condition to be verified before any maximum generator power increase

is that the mean generator power margin has reached a threshold value. That is,

we define the mean generator power margin at a time t as

DP

P

P P e

P e

jj G

t

t=-

Œ

-( )

-( )

 01

01

l

l , (4)

where P0 is the initial power load demand.

(d) Once condition (c) is verified, we choose a node at random to test condition (b). If

the chosen node verifies condition (b), we increase its power by the amount given

by Eq. (3). If condition (b) is not verified, we choose another node at random and

iterate. After power has been added to a node, we use Eq. (4) to recalculate the

mean generator power margin and continue the process until DP/P is above the

prescribed quantity DP Pc( ) .

4. We also assign a probability p0 for a random outage of a line. This value represents

possible failures caused by phenomena such as accidents and weather related events.

After applying these four rules to the network, we look for a solution of the power

flow problem by using linear programming as described in Appendix A.

It is also possible to introduce a time delay between the detection of a limit in the

generation margin and the increase in maximum generator power. This delay would

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represent construction time. However, the result is the same as increasing the value of k

in Eq. (3), which can also give an alternative interpretation for k.

Five basic parameters control the dynamics of this model. One is the rate of increase

in power demand, l, which we keep fixed at 1.8% per year on the basis of the averaged

value for the U.S. grid in the last two decades.22 A second parameter is the improvement

rate of the transmission grid, m. This is not an easy parameter to estimate. However, once

m is given, there is a self-regulation process by which the system produces the number of

blackouts that would stimulate the engineering response needed to meet demand. This is

a necessary condition for the dynamical equilibrium of the system. The rate of increase in

power demand for the overall transmission system is essentially given by R ND Lª -( )l 1 .

The system response is R f NR blackout o Lª -( )m 1 l , where fblackout is the frequency of

blackouts and l o is a weighted average of the number of lines overloaded during a

blackout. Dynamical equilibrium implies that R RD R= . That is, the increase in demand

and the corresponding increase in power supply must be matched by improvements in the

transmission grid. Because those improvements are in response to real or simulated

blackouts, this relation implies that m must be greater than l; otherwise, the system would

be collapsing with constant blackouts. In the numerical calculations and for the value of

the demand increase of 1.8% per year, we found that m must be > 1.01 in order to avoid

this collapse regime. In the present calculations, we keep m in the range 1.01 to 1.1. In

this regime, results depend weakly on m.

A third parameter G is a measure of the generation capacity of the power system in

response to fluctuations in the power demand. G is the ratio of the reserve generator

power to the maximum daily fluctuation of the power demand. The averaged power

demand increases exponentially in time as P t P eDt( ) ∫ -( )

01l . However, the real

instantaneous demand is PD(t), different from the averaged power demand because of

daily fluctuations. The generator power installed PG(t) is also different from the averaged

power demand. The difference DP t P t P tG D( ) ∫ - ( )( ) is the generator capability margin

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used to cope with fluctuations in power demand. In our calculations, the generator

capability margin is varying in time, but we require it to be larger than a minimum

prescribed value DPc . Because the power demand is continuously increasing, it is

convenient to normalize all these quantities to P tD ( ). Thus we define G as the ratio of the

normalized minimal generator capability margin DP P tc D ( ) to the maximum fluctuation

of the load demand g P t P t P tD D D∫ ( ) - ( )( ) ( )[ ]ÊË

ˆ¯Max

2 1 2/

.

G D= [ ]P P gc D , (5)

There is a simple relation between g and the load fluctuation parameter g . The parameter

G is the main parameter varied in the calculations presented here. In the U.S., the

generator power capability margin has had a wide variation over the years, but an

estimated mean value22 falls into the range of 15% to 30%.

The fourth parameter is the probability of an outage caused by a random event (p0).

This parameter can be used to partially control the frequency of blackouts, although the

relation between them is not linear. The fifth parameter is the probability for an

overloaded line to undergo an outage (p1). We keep this parameter in the range

0.1 £ p1 £ 1.0.

Since each calculation can be done for different specific network configurations, in

this work we will use idealized tree-like networks, which were discussed in Ref. 8, as

well as more realistic networks, such as the IEEE 118 bus network depicted in Fig. 1.

The time evolution of a power transmission system represented by this model leads,

after a transient, to a steady-state regime. Here “steady state” is defined with relation to

the slow dynamics of the blackouts because the power demand is constantly increasing,

as shown in Fig. 2. The time evolution in the model shows the transient period followed

by steady-state evolution. This is illustrated in Fig. 2, where we have plotted the number

of blackouts per 300 days as a function of time. We can see a slight increase in the

average number of blackouts during the first 20,000 days. This transient period is

followed by the steady state where the number of blackouts in an averaged sense is

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constant. The properties in the slow transient are not very different from those in the

steady state. However, for statistical analysis, we use the steady-state information to

avoid contamination of the statistics. The length of the transient depends on the rate of

growth in power demand. In the following calculations, we evaluate the blackout

statistics by ignoring the initial transients and doing the calculations for a time period of

80,000 days in a steady state. Of course, the use of these long time scale steady-state

results is driven by the need for large statistical samples and it is arguable whether the

real electric power grid ever actually reaches a steady state.

3. Dynamical evolution of the power transmission model

Looking at the time evolution of the different parameters that characterize the

blackouts, one observes a noisy signal that could be mistaken for random. One could

assume that this is in fact the situation because many of the blackouts are triggered by

random events with probability p0. However, that is not the case. It is instead found that

there are significant space and time correlations resulting from the underlying dynamics

of the power transmission model.

To investigate the time correlations in this apparently noisy system we calculate the

Hurst exponent23 of time series of blackout sizes. Here, we consider two measures of the

size of a blackout. One is the load shed during a blackout normalized to the total power

demand; the other is the number of line outages during a blackout.

We use the R/S method24 to calculate the Hurst exponent. An example of the result of

this analysis is shown in Fig. 3. For times of the order of a few days and a few years,

both series show weak persistence. They have the same Hurst exponent (H = 0.55±0.02).

This result is close to the one obtained in the analysis2 of NERC data on blackouts of the

North American power grid.4 In this range of time scales, the value of the exponent does

not depend on the value of G. For longer times, each time series shows a different

behavior. The load shed has a nearly random character with H = 0.5 for G < 1. For

G > 1, the value of H decreases and in many cases is below 0.5. For these longer time

scales, the time series of the number of line outages has a clear antipersistent character

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with H ranging from 0.2 to 0.4, depending on the network structure. In Fig. 4, the value

of H resulting from a fit of R/S in the time range 600 £ t £ 105 is given as a function of G

for three of the tree networks and for the IEEE 118 bus network. Antipersistency in the

time sequence of number of line outages can be expected from the model. Blackouts with

a large number of line outages happen rarely, only once every few years. When they

happen, there is a great deal of repair and enhancement of many transmission lines. As a

consequence, blackouts with a large number of line outages become less probable after

one of those events. Therefore, there is antipersistency at that time scale. In the present

model, load shed does not have a direct impact on the repair and upgrade of the system.

Therefore, time correlations are weak. As we will discuss in the next section, for G > 1,

blackouts with large load sheds are associated with a large number of line outages.

Therefore, in this G range we see some level of antipersistency due to the coupling of

load shed and the number of line outages. The available data from NERC are limited to

15 years, and we therefore do not have any direct way of confirming this long-term

behavior of the model in the real power system.

The time lag during which the number of line outages changes from weak persistency

to antipersistency is independent of the network size but depends on the repair rate (m).

As m increases, it takes longer time lags for the change to occur. Increasing m causes a

slight increase in H, but H remains less than 0.5.

Within this model the correlations are not limited to time correlations. The PDFs of

the load shed and the number of line outages both have power-scaling regions implying

spatial correlations. The correlations responsible for these power tails are the result of the

system being near a critical point.

In Ref. 7, we studied the critical points of the power transmission model as the total

load demand was varied. The slow dynamics described in Sect. 2 were not modeled. We

found two types of critical points: one type was related to the limiting power flows in the

transmission lines; the other type was related to the limit in the power generation. When

these types of critical points are close to each other, the probability distribution of the

blackout size as measured by the amount of load shed has a power law dependence for a

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range of values of the load shed. Away from the critical point, this power law

dependence no longer exists.

When the dynamical evolution over long time scales is included and the value of G is

about 1, the system naturally evolves to a situation in which these critical points are close

to each other. In this situation, the PDFs of the power shed will have a region of

algebraic decay. In Fig. 5, we have plotted the relative cumulative frequency calculated

from the time series of the blackout data from the numerical results. The cumulative

frequency has been calculated directly from this data using the rank function. In Fig. 5,

the load shed is normalized to the total power demand. The calculation was done for three

of the tree network configurations. These distributions are compared with those obtained

from a load scan without dynamical evolution when the load value was at the critical

point. We cannot distinguish between the two calculations; the relative cumulative

frequencies are practically the same. The overlap between the two results indicates that

the dynamical model described in Sect. 2 intrinsically leads to operation of the system

close to the critical points. A similar result has been obtained for the IEEE 118 bus

network. In Fig. 5, we have given an arbitrary shift to the relative cumulative frequencies

for a given size network to better observe the three different cases.

The relative cumulative frequency plotted in Fig. 5 has three characteristic regions. They

all have an exponential tail reflecting the finite size effect of the network (region III).

Region II is characterized by an algebraic decay. This power-law-scaling region increases

with the number of nodes in the network, suggesting that it is a robust feature of the

system. The power decay index is practically the same for the four networks and is close

to –0.55. The particular values of the decay index for each tree network are given in

Table I, in which the range of the power tail region is defined as the ratio of the

maximum load shed to the minimum load shed described by the power law. From the

values obtained for the four networks listed in Table I, we can see that this range scales

with the network size.

The functional form of the relative cumulative frequency, or at least their power-

scaling region, seems to have a universal character. Therefore, we can compare the

relative cumulative frequency of the normalized load shed obtained for the largest

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network with the relative cumulative frequency of the blackouts obtained in the analysis

of the 15 years of NERC data.4 In Fig. 6, we have plotted the relative cumulative

frequency of the NERC data together with the relative cumulative frequencies for the

382-node tree and IEEE 118 bus networks. We have normalized the blackout size to the

largest blackout over the period of time considered. We can see that the present model,

regardless of the network configuration, reproduces quite well the power-scaling region

from the NERC data. The size of this region is shorter for the calculations. This is

because the calculations are done for relatively small networks. The level of agreement

between the algebraic scaling regions of the relative cumulative frequencies is

remarkable and indicates that the dynamical model for series of blackouts has captured

some of the main features of the NERC data.

4. Dynamical regimes

Calculations carried out with this model show the existence of two different

dynamical regimes. The first regime is characterized by the low value of G (that is, a low

generator capability margin and/or large fluctuations in the power demand). In this

regime, the available power is limited and has difficulties in meeting demand. Blackouts

are frequent, but they affect only a limited number of loads. In this regime, there are very

few line outages. In the opposite limit, G is large and the blackouts are less frequent, but

they tend to involve multiple line outages when they happen. This latter regime is

interesting because there are many cascading events that can cause blackouts in a large

part of the network. This suggests a possible separation between regimes of few failures

and regimes with cascading failures both of which are physically interesting.

Let us investigate in a quantitative way the separation between these two regimes by

varying the parameter G. Varying G is not necessarily a realistic way of modeling the

transmission system but it allows us to understand some features of the dynamics of the

model. For several tree networks, we have done a sequence of calculations for different

values of the minimal generator power margin DP Pc( ) at a constant g . We have

changed this margin from 0 to 100%. For each value of this parameter, we have carried

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out the calculations for more than 100000 days in a steady state regime. One way of

looking at the change of characteristic properties of the blackouts with G is by plotting

the power delivered and the averaged number of line outages per blackout. For a 94-node

tree network, these plots are shown in Fig. 7. We can see that at low and high values of

G, the power served is low. In the first case, because of the limited generator power, the

system cannot deliver enough power when there is a relatively large fluctuation in load

demand. At high G, the power served is low because the number of line outages per

blackout is large.

Looking at averaged quantities is not a good way of identifying the demarcation

between single failures and cascading events. To have a better sense of this demarcation,

we have calculated the PDF of the number of line outages per blackout. In Fig. 8, we

have plotted these PDFs for different values of G. The calculation was done for a 94-

node tree network. We can see that at very low G there is a clear peak at 4 outages per

blackout with very low probability for blackouts with more than 10 outages per blackout.

As G increases, a second peak at about 17 outages emerges and the height of the peak

increases with G. At the highest G value, this second peak is comparable to the peak at

low number outages per blackout. In Fig. 9, we have plotted the ratio of the frequency of

blackouts with more than 15 outages to the mean frequency of blackouts. We can see

that for G > 1, this ratio reaches 0.007. This gives a measure of the frequency of what we

can consider large-scale blackouts (more than 16% of the whole grid). We can apply this

result to the U.S. grid, taking into account that the average frequency of blackouts is one

every 13 days. In the low-G regime, the ratio is about 0.001; this would imply that a

large scale blackout is likely every 35 years. In the high-G regime, the ratio goes up to

0.007; this implies a frequency of one large-scale blackout every 5 years.

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5. Conclusions

The simple mechanisms introduced into the power transmission model and

representing the economical and engineering responses to increasing power demands are

sufficient to introduce a complex behavior in the power system. The results of the

complex dynamics, time correlations, and PDFs of blackout sizes are consistent with the

available data on blackouts of the North American electrical grid.

This model suggests that the real cause of the blackouts in the electric power system

should not be identified just with the immediate random events that trigger them; instead,

the real underlying cause is at a deeper level in the long-term forces that drive the

evolution of the power system.

An important parameter in the system, G , is the ratio of the generator margin

capability to the maximum daily fluctuation of the loads. This is a surrogate for the

systems ability to absorb fluctuations. We do not yet have an economic model for the

time evolution of Gwhich would be the next level of self-consistent evolution for the

system. This parameter allows us to classify the dynamics of the model into two regimes.

At low G, blackouts and brownouts are frequent, and a typical blackout is characterized

by very few line outages. For G > 1, blackouts are less frequent, but large cascading

events involving many line outages are possible.

The dynamical behavior of this model has important implications for power system

planning and operation and for the mitigation of blackout risk. The present model has

some of the characteristic properties of an SOC system, although one cannot

unequivocally prove that is strictly the case. The success of mitigation efforts in complex

systems is strongly influenced by the dynamics of the system. One can understand the

complex dynamics as including opposing forces that drive the system to a “dynamic

equilibrium” near criticality in which disruptions of all sizes occur. Power tails are a

characteristic feature of this dynamic equilibrium. Unless the mitigation efforts alter the

self-organizing dynamical forces driving the system, the system may be pushed toward

criticality. To alter those forces with mitigation efforts may be quite difficult because the

17

forces are an intrinsic part of our society and therefore the power system. Therefore, we

expect that feasible mitigation efforts can move the system to a new dynamic equilibrium

which will remain near criticality and preserve the power tails.5 Thus, while the absolute

frequency of disruptions of all sizes may be reduced, the underlying forces can still cause

the relative frequency of large blackouts to small blackouts to remain the same.

18

Acknowledgments

Ian Dobson and David Newman gratefully acknowledge support in part from NSF

grants ECS-0216053 and ECS-0214369. Ian Dobson and B. A. Carreras gratefully

acknowledge coordination of part of this work by the Consortium for Electric Reliability

Technology Solutions and funding in part by the Assistant Secretary for Energy

Efficiency and Renewable Energy, Office of Power Technologies, Transmission

Reliability Program of the U.S. Department of Energy under contract 9908935 and

Interagency Agreement DE-A1099EE35075 with the National Science Foundation. Part

of this research has been carried out at Oak Ridge National Laboratory, managed by UT-

Battelle, LLC, for the U.S. Department of Energy under contract number DE-AC05-

00OR22725.

19

Appendix A

The blackout model is based on the standard DC power flow equation,

F AP= , (A-1)

where F is a vector whose NL components are the power flows through the lines, Fij, P is

a vector whose NN–1 components are the power of each node, Pi, with the exception of

the reference generator, P0, and A is a constant matrix. The reference generator power is

not included in the vector P to avoid singularity of A as a consequence of the overall

power balance.

The input power demands are either specified deterministically or as an average value

plus some random fluctuation around the average value. The random fluctuation is

applied to either each load or to “regional” groups of load nodes.

The generator power dispatch is solved using standard LP methods. Using the input

power demand, we solve the power flow equations, Eq. (A-1), with the condition of

minimizing the following cost function:

Cost = ( ) - ( )Œ Œ

 ÂP t W P tii G

jj L

. (A-2)

We assume that all generators run at the same cost and that all loads have the same

priority to be served. However, we set up a high price for load shed by setting W at 100.

This minimization is done with the following constraints:

(1) Generator power 0 £ £ ŒP P i Gi imax

(2) Load power P j Lj £ Œ0

(3) Power flows F Fij ij£ max

(4) Power balance Pii G LŒ »

 = 0

20

This linear programming problem is numerically solved by using the simplex method

as implemented in.25 The assumption of uniform cost and load priority can of course be

relaxed, but changes to the underlying dynamics are not likely from this.

In solving the power dispatch problem for low-load power demands, the initial

conditions are chosen in such a way that a feasible solution of the linear programming

problem exists. That is, the initial conditions yield a solution without line overloads and

without power shed. Increases in the average load powers and random load fluctuations

can cause a solution of the linear programming with line overloads or requiring load

power to be shed. At this point, a cascading event may be triggered.

A cascading overload may start if one or more lines are overloaded in the solution of

the linear programming problem. We consider a line to be overloaded if the power flow

through it is within 1% of Fijmax . At this point, we assume that there is a probability p1

that an overloaded line will cause a line outage. If an overloaded line experiences an

outage, we reduce its corresponding Fijmax by a large amount (making it effectively zero)

to simulate the outage, and calculate a new solution. This process can require multiple

iterations and continues until a solution is found with no more outages.

This fast dynamics model does not attempt to capture the intricate details of particular

blackouts, which may have a large variety of complicated interacting processes also

involving, for example, protection systems, and dynamics and human factors. However,

the fast dynamics model does represent cascading overloads and outages that are

consistent with some basic network and operational constraints.

21

Table I

Number of nodes PDF decay index Range of power tail

46 -0.56 494 -0.51 8190 -0.55 13382 -0.58 31

22

References

1D. N. Ewart, IEEE Spectrum, 36 (1978).2B. A. Carreras, D. E. Newman, I. Dobson, and A. B. Poole, “Evidence for self-

organized criticality in electric power system blackouts,” paper submitted to IEEE

Transactions on Circuits and Systems Part I .3J. Chen, J. S. Thorp, and M. Parashar, “Analysis of electric power disturbance data,”

34th Hawaii International Conference on System Sciences, Maui, Hawaii, January

2001.4Information on electric systems disturbances in North America can be downloaded

from the NERC website at http://www.nerc.com/dawg/database.html.5B. A. Carreras, V. Lynch, I. Dobson, and D. E. Newman, “Blackout mitigation

assessment in power transmission systems,” 36th Hawaii International Conference on

System Sciences, Hawaii, January 2003. Available from IEEE at

http://ieeexplore.ieee.org.6I. Dobson, B. A. Carreras, V. Lynch, and D. E. Newman, “An initial model for complex

dynamics in electric power system blackouts,” 34th Hawaii International Conference on

System Sciences, Maui, Hawaii, January 2001. Available from IEEE at

http://ieeexplore.ieee.org.7B. A. Carreras, D. E. Newman, I. Dobson, and A. B. Poole, “Modeling blackout

dynamics in power transmission networks with simple structure,” 34th Hawaii

International Conference on System Sciences, Maui, Hawaii, January 2001. Available

from IEEE at http://ieeexplore.ieee.org.8B. A. Carreras, V. Lynch, I. Dobson, and D. E. Newman, CHAOS, 12, 985 (2002).9P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 1987, pp. 381-4.

10M. L. Sachtjen, B. A. Carreras, and V. E. Lynch, Phys. Rev. E 61, 4877 (2000).11J. Chen and J. S. Thorp, “A reliability study of transmission system protection via a

hidden failure DC load flow model,” IEEE Fifth International Conference on Power

System Management and Control, 17-19 April 2002, 384-389.

12J. Chen, J. S. Thorp, and I. Dobson, “Cascading dynamics and mitigation assessment in

power system disturbances via a hidden failure model,” preprint, submitted to

23

International Journal of Electrical Power and Energy Systems, 2003.13M. D. Stubna and J. Fowler, International Journal of Bifurcation and Chaos, 13 (1), 237

(2003).14 S. Roy, C. Asavathiratham, B. C. Lesieutre, and G. C. Verghese, “Network models:

growth, dynamics, and failure,” Proceedings of the 34th Annual Hawaii International

Conference on System Sciences, January 3-6, 2001, pp. 728 –737.

15D. L. Pepyne, C. G. Panayiotou, C. G. Cassandras, and Y.-C. Ho, “Vulnerability

assessment and allocation of protection resources in power systems,” Proceedings of

the American Control Conference, Vol. 6 , 25-27 June 2001, pp. 4705 –4710.

16C. L. DeMarco, “A phase transition model for cascading network failure,” IEEE

Control Systems Magazine, Vol. 21, No. 6, pp. 40-51, December 2001.

17P. A. Parrilo, S. Lall, F. Paganini, G. C. Verghese, B. C. Lesieutre, J. E. Marsden,

“Model reduction for analysis of cascading failures in power systems,” Proceedings of

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18B. Stott and E. Hobson, IEEE Trans. Power Appar. Syst., PAS-97, 1713 (1978).19B. Stott and E. Hobson, IEEE Trans. Power Appar. Syst., PAS-97, 1721(1978).20B. Stott and J. L. Marinho, IEEE Trans. Power Appar. Syst., PAS-97, 837 (1979).2 1 The IEEE 118 bus network model is a standard test system; see

http://www.ee.washington.edu/research/pstca/.22Statistical Yearbook of the electric utility industry/1998, published by Edison Electric

Institute (1998).23H. E. Hurst, Trans. Am. Soc. Civil Eng. 116, 770 (1951).24B. B. Mandelbrot and J. R. Wallis, Water Resources Research 4, 909–918 (1969).25W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes

in C (Cambridge University Press, Cambridge, 1988).

24

Figure captions

Fig. 1. Diagram of the IEEE 118 bus network. Generators are gray squares; loads are

the black squares.

Fig. 2. Time evolution of the power served and number of blackouts per year from the

model.

Fig. 3. R/S for the time series of normalized load shed and line outages for a 46-node

tree network.

Fig. 4. The Hurst exponent H as a function of G for the time series of normalized load

shed (a) and line outages (b). The exponent is calculated from a fit of R/S in

the time range 600 £ t £ 105 for 46, 94, and 190 nodes tree networks and for

the IEEE 118 bus network.

Fig. 5. Relative cumulative frequency of the load shed normalized to the total power

demand for three different tree networks. The relative cumulative frequencies

obtained from a load scan near the critical point are compared with the relative

cumulative frequencies obtained from the dynamical model discussed in this

paper.

Fig. 6. Relative cumulative frequencies of the normalized load shed for the 382-node

tree, the IEEE 118 bus networks, and the North American blackouts in 15

years of NERC data normalized to the largest blackout.

Fig. 7: Averaged power delivered and number of line outages per blackout for the 94-

node tree network as a function of G.

Fig. 8: PDF of the number of outages per blackout for the 94-node tree network for

different values of G.

Fig. 9: Ratio of the frequency of blackouts with more than 15 outages to the frequency

of blackouts for the 94-node tree network as a function of G.

25

Fig. 1

26

Fig. 2

27

10-1

100

101

102

103

100 101 102 103 104 105

R/S

Time lag

Power shed/PS

Line outages

Fig. 3

28

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Tree 46 nodes

Tree 94 nodes

Tree 190 nodes

IEEE 118 bus

H

G

a)

Fig. 4a

29

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Tree 46 nodesTree 94 nodesTree 190 nodesIEEE 118 bus

H

G

b)

H = 0.5

Fig. 4b

30

10-2

10-1

100

101

102

10-3 10-2 10-1 100

N=46N=46 Dynamical modelN=94N=94 Dynamical modelN=190N=190 Dynamical model

Rel

ativ

e cu

mul

ativ

e fr

eque

ncy

Load Shed/Power Demand

I II III

Fig. 5

31

10-3

10-2

10-1

100

101

10-4 10-3 10-2 10-1 100

U.S. grid data382-node NetworkIEEE 118 bus NetworkR

elat

ive

cum

ulat

ive

freq

uenc

y

Load shed/Power demand

Fig. 6

32

Fig. 7

33

Fig. 8

34

Fig. 9